Question

Electric circuits having an inductor, resistor, capacitor and an electromotive force connected in series (RLC circuits) are modeled

by a differential equation analogous to the one that models the mass-spring oscillator. If the inductance is L (henrys), the resistance
is R (ohms), the capacitance is C (farads) and the electromotive force is E(t) (volts), the differential equation governing the charge
q (coulombs) is
Lq+Rq+1/C q = E(t)
with the initial conditions given by the initial charge, q(0), and the initial current i(0) in the circuit. Consider a circuit
containing an inductor Of I henry, a resistor Of 100 Ohms, a capacitor Of 10¯4 farads driven by an electromotive force Of E(t)
962 sin 60t volts.
(a) Write the differential equation governing the charge in the circuit.
(b) Find the general solution Of the differential equation in part (a).
(c) For the differential equation in part (a), find the particular solution to the initial value problem if there is no current and no
charge in the circuit at time t — 0.
(d) Find the value of the steady state charge and steady state current (i(t) = when t = 7/60 seconds.

Answer 1

The differential equation governing the charge in an RLC circuit is Lq + Rq + 1/Cq = E(t). The general solution of the differential equation is q(t) = A cos(wt + phi). The particular solution to the initial value problem can be found by substituting the initial conditions into the general solution.

Step-by-step explanation:

The differential equation governing the charge in the RLC circuit is Lq + Rq + 1/Cq = E(t). In this case, the inductance is I henry, the resistance is 100 ohms, the capacitance is 10^-4 farads, and the electromotive force is given by E(t) = 962 sin(60t) volts.

The general solution of the differential equation is q(t) = A cos(wt + phi), where A and phi are constants and w is the angular frequency given by sqrt(IC).

The particular solution to the initial value problem, where there is no current and no charge in the circuit at t = 0, can be found by substituting the initial conditions into the general solution.

The value of the steady state charge and steady state current at t = 7/60 seconds can be found by substituting t = 7/60 into the particular solution.